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Optimizing and satisficing
Structural Safety, 7 (1990) 155-163
155
Elsevier
OPTIMIZING
AND SATISFICING
*
Colin B. Brown Department of Civil Engineering, University of Washington, Seattle, WA 98195 (U.S.A.)
(Accepted May 1989)
Key words: design; optimization; satisficing; uncertainty; fuzzy sets; probabilities.
ABSTRACT
Optimization has the appeal of mathematical elegance and the possibility of being of practical significance. This significance is examined for the problem of producing least-cost structures. For aircraft, this simplifies to the production of structures of least weight, whereas in civil engineering, the existing conditions for design and construction in the United States seldom produce an environment for the existence of least-cost structures. Optimization procedures are examined within these situations when the objective functions and constraints are deterministic and when they are uncertain. The uncertainty is expressed as amenable to both probabilistic and fuzzy set specifications. This transformation, from crisp optimization to the concept of satisficing of structures, is explored.
1. INTRODUCTION
The process of optimism has the dual of pessimism and an osculation of these two states, which is the mini-max solution. This is the best of all possible worlds of the optimistic Dr. Pangloss in Voltaire's Candide and leads to J.B. Cabell's use of the literal definition of the words as: "The optimist proclaims that we live in the best of all possible worlds; and the pessimist fears this is true." The process of optimization seeks the world of the optimist by means of determining the decision variables, x ~ X, the decision space, such that the objective function, f ( x ) , is an extremum in the feasible space of all x in G, the constraints. More formally we seek the value J where J=inf{f(x);
x~G),
G~X
(1)
* Presented at the Workshop on Research Needs for Applications of System Reliability Concepts and Techniques in Structural Analysis, Design and Optimization, Boulder, CO, September 12-14, 1988.
156 and f is a real-valued function. This can be generalized when f(x) is a multi-dimensional vector and simplified when the objective function and constraints have special forms, such as linear. If it exists, then the minimum rather than the infimum of f can be determined in eqn. (1). In structural optimization J is the minimum cost structure subject to safety constraints; in the design of aircraft this is treated as synonymous with the minimum weight structure. The sense in this change is found in the dynamic design of aircraft whereby F = m • a, with a as the kinematic specification and F as the thrust necessary to attain it. Then the minimization of the weight, or m, also minimizes the thrust F for a required performance, a. The cost of the aircraft is a combination of the costs of the thrust units and the body weight; hence, a minimization of weight provides a minimization of cost. This same fortunate argument cannot be applied to civil engineering structures where the problem is treated as pseudo-static and F = 0. In this situation an increase in weight does not necessarily result in an increase in cost; hence, a weight minimization may not be an appropriate optimization. The practical selection of a minimum cost structure requires the bidding by competing contractors on a design supplied by the owner's representative. The design meets the intentions of the owner and is the instrument for obtaining cost bids. Such a design must attract enough bidders to ensure competition on the basis of cost and therefore must not include features that would favor one bidder and exclude others. This process usually guarantees that the successful bid is not one of minimum cost; a design which utilized the particular skills and equipment of that or one of the other contractors would usually result in a lower cost. Collusion in this form is encouraged in other parts of the world, but not in North America. It may lead to the minimum cost structure if, and only if, that structure is included among the competing alternatives. A solution of eqn. (1) produces a design that is the best among the alternatives considered, subject to the constraints considered and guided by the objective function considered. These alternatives, constraints and objectives are limited by the human mind; they may, in fact, be an incomplete model of the real world. In all signifcant practical problems the human mind fails in the modeling of the actual reality; it is this inability to capture and solve real, complex problems that suggested to Simon the necessity of a Principle of Bounded Rationality [1]. In this principle, formal solutions to eqn. (1), in which the alternatives, the constraints and the objectives are incomplete, cannot produce global answers. In this, Simon separates the limited rationality employed by man from the global rationality necessary to attain a complete solution to a real problem. The limited rationality is what professionals use when they seek solutions to real problems in a finite time in the face of uncertainty about the world. Such a definition of a professional is fitting; engineers, lawyers, doctors and soldiers would find it applicable. Based upon this Principle, Simon [2] urged decision-makers to abandon optimization and instead seek solutions which are acceptable. He termed the process whereby the acceptable, rather than the best of all possible worlds, is sought as satisficing. Certainly in complicated problems of structural design the conditions for the Principle of Bounded Rationality exist and satisficing would seem to be appropriate. Much that goes on in a civilized society involves optimizing in the form of eqn. (1). Mannheim [3] labels this as functional rationality. He separates this from substantial rationality which reveals insight into causative and correlated events which may not be calculable. The claim is that most people in civilized society are involved with algorithms such as eqn. (1), and only a few adopt substantial rationality with the consequent necessity of independent thinking. Mannheim appears to be concerned with sociological classifications in which both optimizing and satisficing are grouped as functional rationality. Simon's interest is in the effectiveness of
157 rational procedures and he finds that the time and costs of acquiring the information necessary to optimize are professionally prohibitive and hence suggests the viability of satisficing. In this paper, aspects of optimization that may concern structural engineers are considered. These include true costs, minimum weight strategies, holistic systems and uncertainty. Such considerations are discussed in the light of the comments in this Introduction and lead to the associated problem of satisficing.
2. COSTS The initial cost of a structure must be included in the total cost. Additional to this is the design expense which differs with the type of structure considered. Design costs are related to length of time spent on the process. Table 1 illustrates the time in man-years spent for engineering various types of structures. With the exception of spacecraft, the design costs are related to the fleet size, or number of replicates, and structural nakedness. They increase as the number of similar articles intended increases and as the critical nature of the structural form is emphasized. Thus, novel structures, bridges and ships, which have critical and clear structural forms, have high design times, whereas buildings, which have hidden strength from non-structural parts have lower values. The initial cost, C, includes the actual cost of the structure, design and any insurance provided. Such insurance depends on the chances of failure of the designed structure, p, and hence C = C(p). D is the replacement cost and E is the cost associated with failure due to errors. The chance of this is P and hence E = E(P). A statement of the objective function of costs to be minimized in eqn. (1) is
J(x; p, P)=min[C(x; p)+D(p+P)+E(P).P]
(2)
X
where x are the constrained variables of geometry and weight. The dependence of costs upon x, p and P is clear. However, p is tied to x. Therefore, in the extremum problem of eqn. (2), the x solution to minimize J for prescribed levels of safety, p, and error control, P, is sought. This is satisfactory in a formal sense, but at this time there is not a proven relationship between error control and P. Of concern in eqn. (2) is the error costs, E; these may be very small to the owner if dealt with by insurance premiums. Then, E << C + D and eqn. (2) can be thought of as
J(x; p, P)=min[C(x; p)+D(p+e)]
(3)
X
Decisions on safety and design and construction control establish p and P, leaving x as the optimization variables. If the costs are considered in a more general sense, then E may dominate
TABLE 1 Comparison of the design time per ton of structure of various structural types Type of structure Spacecraft Aircraft Offshore structures: Novel Conventional Bridges and ships Buildings
Engineering man-years 500 10 0.01 0.001 0.0006 0.0002
158
over C and D. Also, the computed probability of failure, p, is often orders of magnitude smaller than the failure incidence [4]. The difference may be accounted for by design-construction-operation errors and hence p << P. These comments suggest that eqn. (3) is often appropriate. Additional to the features in eqn. (2) are costs of the operation of the structure. These, like any long-term costs, have to account for future inflation and interest rates in order to use common initial costs in the objective function. All these pricings, including the elements of eqn. (2), have to be obtained a priori; the usual process is to maintain consistency in assessing each cost alternative. These adjustments amount to exercises in functional and bounded rationality in the senses of Mannheim and Simon.
3. MINIMUM WEIGHT DESIGN The use of weight as an objective function is more direct and simpler than cost. Unfortunately, except in dynamic design, it is a less satisfactory vicarious form. If there are n elements each of weight x(i), then the mathematical program is Y/
rain E x( i )
(4)
i=1
subject to
x(i)
i=1 ton
stress safety constraints displacement constraints stability safety constraints frequency constraints
(5a) (5b) (5c) (5d) (5e)
are prescribed bounds on x. The constraints of eqns. (5b) and (5c) require the Here x and completion of the structural analysis problems K.r--Q
(6)
for each loading and each alternative in the path towards optimality. The constraints of eqns. (5d) and (5e) require the completion of the characteristic value problems
(K-X.M).r=O
(7)
where M is the geometric stiffness matrix for the stability case and is the mass matrix for the frequency one. The resulting stresses and displacements from eqn. (6) are constrained to be no greater than limiting values for all loadings. The actual loads are constrained to be smaller than the factored lowest characteristic loads from eqn. (7) to ensure member stability; the frequencies from eqn. (7) are constrained to avoid specified values. The constraints of eqn. (5a) limit the admissible designs to available member sizes. Search procedures have been developed to determine the optimum variables, x*(i), of the mathematical program of eqns. (4) and (5). These are computationally intensive and require the re-evaluation of eqns. (6) and (7) for each design alternative. Only in fleet or completely novel structures are such search procedures justified. More often the approach is changed to seek the design whereby all members are maximally stressed under at least one loading state. Thus, the optimality criterion of the fully stressed design becomes the surrogate for the minimum weight
159 design. The program search is replaced by a recursion relationship and the constraints of eqns. (5c), (5d) and (5e) are checked on the fully stressed candidate. The mathematical program involving the objective function and constraints may be written as a Lagrangian with the same number of multipliers as the constraints. Such a Lagrangian, L(x, ~), with multipliers, 3'j, is
L(x, g)=f(x)+
Y'~yj{Cj(x)}
(8)
J
with Cj(x) as the homogeneous constraints. A constrained local extremum on x = x* subject to satisfying the K a r u s h - K u h n - T u c k e r conditions.
f(x)
exists at
4. STRUCTURAL SYSTEMS The form of eqn. (1) may apply not only to members but also to structural systems, which are comprised of members with appropriate connectiv!ty and boundary conditions. The constraints for a linear structural system require the solution of eqns. (6) and (7), whether the objective function is for cost or weight. Such a linear structure, with strains restrained to limited elastic response, are applicable to the functional attitude to design. Here failure is specified as a stress, load, displacement or frequency level or range, and such states are obtained by a linear approach. This is usual in space and air structures and for off-shore and ship structures. In buildings and bridges, failure is considered as collapse, and optimization work has been applied to plastic analysis whereby eqn. (6) is replaced by linear programs where the K a r u s h - K u h n - T u c k e r conditions are necessary and sufficient, and where the multipliers, yj, in the dual lead to physical interpretation and a static-kinematic duality [5,6]. This approach has been extended to plastic limit synthesis with weight as the objective function [7]. The computer programming activity in the optimation of structural systems, even when p and P are zero, has been extensive. With adequate resources, such m i n i m u m weight structures can be attained. Interestingly, the iterative process of conventional design, whereby a solution is modified in the light of the proximity to the constraints until a design in which the constraints are satisfied results, has been shown to approach a minimum weight form [8].
5. UNCERTAINTY The statement of eqns. (2) and (3) has uncertainty defined in a frequentist sense evident in p, the probability that the structure will fail when properly designed, constructed and maintained; and P, the probability of errors increasing the failure rate to p + P. Also, the constraints of eqn. (5) have uncertainties imposed in a rational, functional sense inasmuch as the stress and load constraints have a formalized ritual of Level 1 or, at the best, Level 2 methods of assessing structural safety. The significance of these two interpretations of uncertainty in the cost objective function and the safety constraints is deeper than a violation of rational consistency. It is an example of Simon's Principle of Bounded Rationality; either the objectives or the constraints, or both, are incomplete, and the solutions that are forthcoming are not global ones. To illustrate this state consider the objective statement of eqn. (2) where x, p and P are optimization variables. In an
160 unconstrained procedure the design, x, will have minimum costs at safety levels, p and P, which may be unacceptable. The constraints of eqns. (5b) and (5d) are intended to ensure appropriate safety in the collapse sense. These constraints should provide upper limits on P and p, which establish ranges for them in eqn. (2) and hence result in another design, x, different from x, with adequate safety levels, P and p. Such an approach requires that the measures of uncertainties, p and P, are the same in both the objective function and the safety constraints. However, if different measures are used in the cost objective and the safety constraint, then another solution, x, will be found in which no definite statements can be made on the uncertainties p and P. What can be stated are values of p and P on the costs and /3 on the safety. Relationships between /3, the reliability index, and p + P can be made only if p + P have particular forms, such as a normal distribution. Certainly, a satisfactory algorithm exists in this mixed optimization which has functional rationality. However, the meaning is not that of the best of all worlds but an artificial solution of doubtful merit. The expression of the uncertainty as probabilities with a frequentist interpretation in eqns. (2) and (3) is essential in ascertaining costs. The conflict with the usual Bayesian probability in eqns. (5b) and (5d) can be resolved by changing from a cost to a weight objective function. Then objective uncertainty is largely removed, and the subjective uncertainty in eqns. (5b) and (5d) is left. Some concern must still be expressed inasmuch as the minimum weight structure so obtained accounts for realistic uncertainty only an a vicarious /3 form. The use of optimality critera, which is the more normal procedure, to replace cost and weight as objectives suffers from the same absence of realism. However, once again an algorithm with associated functional rationality exists.
6. SATISFICING The Introduction displayed reasons for adopting a satisficing approach to structural design based upon the bounded rationality of Simon and the substantial rationality of Mannheim. Additionally, concern about the statement of costs and the dichotomy in the definitions of failure adopted in the safety constraints and in the objective function tend to suggest a less optimistic and crisp approach to design. The complex nature of a contemporary structure requires sophisticated analysis in the form of eqns. (6) and (7) at each stage in the approach towards a formal optimum. Essentially, the optimization procedure is demanding high precision about a complex system. Zadeh [9] had something to say on this matter in his Principle of Incompatibility: " . . . as the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics." In structural optimization procedures, precision on uncertainty, member location and connectivity, material parameters, and loading and boundary conditions is essential; the analytical tools are an example of precision; the best of all worlds that the solution provides is a precise answer of doubtful significance. There are other, softer ways of proceeding. Roy [10] has provided three approaches relevant to eqn. (1): (1) determine the unique and best solution from x;
161 (2) determine all solutions x ~ X that are "good" and from these consider further to determine those that are equivalently "satisfactory"; (3) determine classes of solutions x ~ X, each of which is equivalent in value, and construct an ordered sequence of these indifference classes. The first approach is optimization; the second two in the end determine alternatives which are of equal value or, more usefully, between which all parties are indifferent. This addresses one aspect of satisficing: the detection of a class of solutions to which indifference between the members of the class exists. The indifference measures need not be precise; they may in fact be vague, but there must be agreement that they evaluate the required structural characteristics of cost, safety and performance in a way that does not allow the actors to prefer one member in an indifference class over another. Such an approach is valid when the constraints are not realistically precise and where the objectives are satisfied by various criteria. A corollary to Zadeh's Principle of Incompatibility has been expressed as: "The closer one looks at a real world problem, the fuzzier becomes its solution." It is this fuzziness or vagueness that can be reflected in approaches (1) and (2) above and which has led to fuzzy decision-making. In this topic, the formalities of mathematical programming of optimization have been mirrored into fuzzy mathematical programming where the objectives and constraints may be fuzzy or where, therefore, the vague membership in a class or set can be dealt with by fuzzy set theory. This has been reviewed recently by Zimmermann [11]. A real complication exists with such an approach inasmuch as the p and P in eqns. (2) and (3) and the fl in the safety constraints are not vague; they are crisp numbers which in the case of p and P are definitely about crisp sets. The fuzzy programming has much the same operational procedures as goal programming and is efficient in obtaining solutions. Essentially, the approach (1) above is being fuzzified. Approaches (2) and (3) rely on ordering and clustering to approach a "best" indifference class. The selection among the alternatives requires further analysis by the comparison of the m goals of the objective function applied to each alternative. Saaty [12] considered this problem from a psychological viewpoint and concluded that reductionism was superior to aggregation. Rather than comparing all alternatives with respect to the composite of goals, he compared alternatives with respect to each goal (A) on a scale of zero to unity. The weighting among the goals involved a characteristic value problem in which the m weights of pairs of goals are determined and assembled into a diagonally inverse m X m matrix. The characteristic vector from the matrix became the weighting vector which then was used to modify the matrix A to obtain a final comparison matrix of the goals for each alternative. The maxi-min alternative provided the decision case. Such an approach, although formal, has the feature of reliablity embedded into it. It appeals to pairwise comparisons and simple algebraic operations to obtain a final sense between the members of the indifference class. It also deals with the multi-objective criteria in crisper ways than when an attemp to compare directly the alternative ranking for all goals. These procedures allow for satisficing among competing alternatives. Another aspect is that the selected alternative must be robust with respect to a range of changes in the parameters, models, actors and uses that are realistically likely to occur. Geometrically, this means that the extremum must be flat and broad in the variable space. Testing for such robustness can be conducted by sensitivity analysis and by indifference to change in the satisficing procedure.
162
7. C O M M E N T S
The previous discussion displays a separation between optimizing and satisficing. Optimizing is a procedure based on strong mathematical foundations which allows precise statements to be made in the objective function, constraints and solution. Satisficing is an expectation with little formal trappings which admits of imprecision in the objectives and constraints and robustness in the solution. Applications of optimization in design requires an act of confidence in the procedures which can seldom be supported by the reality of the problem. Applications of satisficing reveal the reality with confidence but are based upon undeveloped procedures. It would appear that some combination of the two would indeed be the best of all possible worlds. However, at the moment optimizing is an act of functional and limited rationality; satisficing has the appearance of global rationality and respects the Principle of Bounded Rationality. Certainly an optimization scheme ignores the Principal of Incompatibility; satisficing may respect the Principle. These comments indicate that future research should attempt to introduce realism into the optimization scheme but, in the process, should anticipate a reduced precision in objective, constraints and solution. The measures for such studies should be the extent that realism co-exists with rigorous analytics. Cooperation between practitioners and researchers seems inevitable if realism is to be tied to what goes on in practice. Indeed, studies that probe the extent that present-day design is a satisficing procedure with convergent and robust methodologies should be examined. Spillers [8] has already given some insight into this approach. Uncertainty will always exist in design of complex structures and Zadeh [9] has suggested that added precision does little or nothing to ameliorate it. The use of probability in a frequentist sense seems to be inevitable in describing costs and should be evident in constraint statements. However, at present a confusion exists whereby constraints are expressed by Bayesian surrogates and costs by implied frequencies. In addition, any move away from precision in the sense of Zadeh [9] calls for measures of imprecision. In this he has advocated the use of fuzzy sets. Some merging must occur between objective and subjective probabilities and fuzzy sets. At the moment there are clear separations in the axioms of probability and fuzzy sets which can be formally studied. However, the meanings of objective and subjective probabilities are dramatically different and lend to confusion when used separately in objectives and constraints.
8. C O N C L U S I O N
An attempt has been made to examine the features of optimizing and satisficing. Optimization is a mathematical activity and can be formally described. The extent that the formalism cannot ape reality has been explored under the umbrella of the principles of Simon [1], Mannheim [3] and Zadeh [9]. These shortcomings suggest another approach to structural design which is easy to appreciate but, as yet, is not established in a formal manner. However, studies can be organized which, in the end, may make use of the formal advantages of optimization and the realism of satisficing. Possible forms of such an hybrid are discussed in the paper.
ACKNOWLEDGEMENT This work was supported by the National Science Foundation, Grant No. ECE 8518155
163
REFERENCES 1 H.A. Simon, Models of Man, Social and Rational: Mathematical Essays on Rational Human Behavior in Social Settings, Wiley, New York, 1957. 2 H.A. Simon, Administrative Behavior, 2nd edn., Free Press, New York, 1957, Introduction. 3 K. Mannheim, Man and Society in an Age of Reconstruction: Studies in Modern Social Structure, Kegan Paul, Trench, Trubner, London, 1940. 4 C.B. Brown, A fuzzy safety measure, J. Eng. Mech. Div., ASCE, 105 (EM5) (1979) 855-872. 5 G. Maier and J. Munro, Mathematical programming applications to engineering plastic analysis, Appl. Mech. Rev., 35 (2) (1982) 1631-1643. 6 G. Maier and D. Lloyd Smith, Update to "Mathematical programming applications to engineering plastic analysis", Appl. Mech. Rev. Update, (1986) 377. 7 J. Munro, Optimal plastic design, NATO-ASI Lecture Notes, Engineering Plasticity by Mathematical Programming, Waterloo, Ontario, 1977. 8 W.R. Spillers and J. Farrell, On the analysis of structural design, J. Math. Anal. Appl., 25 (2) (1969). 9 L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems, Man and Cybernetics, SMC-3 (1) (1973) 28-44. 10 B. Roy, Problems and methods with multiple objective functions, Math. Prog. 1 (1971) 239-266. 11 H.J. Zimmermann, Fuzzy Sets, Decision Making and Expert Systems, Kluwer Academic Publishers, Boston, 1987. 12 T.L. Saaty, A scaling for priorities in hierarchial structures, J. Math. Psychol., 15 (1977) 234-281.
155
Elsevier
OPTIMIZING
AND SATISFICING
*
Colin B. Brown Department of Civil Engineering, University of Washington, Seattle, WA 98195 (U.S.A.)
(Accepted May 1989)
Key words: design; optimization; satisficing; uncertainty; fuzzy sets; probabilities.
ABSTRACT
Optimization has the appeal of mathematical elegance and the possibility of being of practical significance. This significance is examined for the problem of producing least-cost structures. For aircraft, this simplifies to the production of structures of least weight, whereas in civil engineering, the existing conditions for design and construction in the United States seldom produce an environment for the existence of least-cost structures. Optimization procedures are examined within these situations when the objective functions and constraints are deterministic and when they are uncertain. The uncertainty is expressed as amenable to both probabilistic and fuzzy set specifications. This transformation, from crisp optimization to the concept of satisficing of structures, is explored.
1. INTRODUCTION
The process of optimism has the dual of pessimism and an osculation of these two states, which is the mini-max solution. This is the best of all possible worlds of the optimistic Dr. Pangloss in Voltaire's Candide and leads to J.B. Cabell's use of the literal definition of the words as: "The optimist proclaims that we live in the best of all possible worlds; and the pessimist fears this is true." The process of optimization seeks the world of the optimist by means of determining the decision variables, x ~ X, the decision space, such that the objective function, f ( x ) , is an extremum in the feasible space of all x in G, the constraints. More formally we seek the value J where J=inf{f(x);
x~G),
G~X
(1)
* Presented at the Workshop on Research Needs for Applications of System Reliability Concepts and Techniques in Structural Analysis, Design and Optimization, Boulder, CO, September 12-14, 1988.
156 and f is a real-valued function. This can be generalized when f(x) is a multi-dimensional vector and simplified when the objective function and constraints have special forms, such as linear. If it exists, then the minimum rather than the infimum of f can be determined in eqn. (1). In structural optimization J is the minimum cost structure subject to safety constraints; in the design of aircraft this is treated as synonymous with the minimum weight structure. The sense in this change is found in the dynamic design of aircraft whereby F = m • a, with a as the kinematic specification and F as the thrust necessary to attain it. Then the minimization of the weight, or m, also minimizes the thrust F for a required performance, a. The cost of the aircraft is a combination of the costs of the thrust units and the body weight; hence, a minimization of weight provides a minimization of cost. This same fortunate argument cannot be applied to civil engineering structures where the problem is treated as pseudo-static and F = 0. In this situation an increase in weight does not necessarily result in an increase in cost; hence, a weight minimization may not be an appropriate optimization. The practical selection of a minimum cost structure requires the bidding by competing contractors on a design supplied by the owner's representative. The design meets the intentions of the owner and is the instrument for obtaining cost bids. Such a design must attract enough bidders to ensure competition on the basis of cost and therefore must not include features that would favor one bidder and exclude others. This process usually guarantees that the successful bid is not one of minimum cost; a design which utilized the particular skills and equipment of that or one of the other contractors would usually result in a lower cost. Collusion in this form is encouraged in other parts of the world, but not in North America. It may lead to the minimum cost structure if, and only if, that structure is included among the competing alternatives. A solution of eqn. (1) produces a design that is the best among the alternatives considered, subject to the constraints considered and guided by the objective function considered. These alternatives, constraints and objectives are limited by the human mind; they may, in fact, be an incomplete model of the real world. In all signifcant practical problems the human mind fails in the modeling of the actual reality; it is this inability to capture and solve real, complex problems that suggested to Simon the necessity of a Principle of Bounded Rationality [1]. In this principle, formal solutions to eqn. (1), in which the alternatives, the constraints and the objectives are incomplete, cannot produce global answers. In this, Simon separates the limited rationality employed by man from the global rationality necessary to attain a complete solution to a real problem. The limited rationality is what professionals use when they seek solutions to real problems in a finite time in the face of uncertainty about the world. Such a definition of a professional is fitting; engineers, lawyers, doctors and soldiers would find it applicable. Based upon this Principle, Simon [2] urged decision-makers to abandon optimization and instead seek solutions which are acceptable. He termed the process whereby the acceptable, rather than the best of all possible worlds, is sought as satisficing. Certainly in complicated problems of structural design the conditions for the Principle of Bounded Rationality exist and satisficing would seem to be appropriate. Much that goes on in a civilized society involves optimizing in the form of eqn. (1). Mannheim [3] labels this as functional rationality. He separates this from substantial rationality which reveals insight into causative and correlated events which may not be calculable. The claim is that most people in civilized society are involved with algorithms such as eqn. (1), and only a few adopt substantial rationality with the consequent necessity of independent thinking. Mannheim appears to be concerned with sociological classifications in which both optimizing and satisficing are grouped as functional rationality. Simon's interest is in the effectiveness of
157 rational procedures and he finds that the time and costs of acquiring the information necessary to optimize are professionally prohibitive and hence suggests the viability of satisficing. In this paper, aspects of optimization that may concern structural engineers are considered. These include true costs, minimum weight strategies, holistic systems and uncertainty. Such considerations are discussed in the light of the comments in this Introduction and lead to the associated problem of satisficing.
2. COSTS The initial cost of a structure must be included in the total cost. Additional to this is the design expense which differs with the type of structure considered. Design costs are related to length of time spent on the process. Table 1 illustrates the time in man-years spent for engineering various types of structures. With the exception of spacecraft, the design costs are related to the fleet size, or number of replicates, and structural nakedness. They increase as the number of similar articles intended increases and as the critical nature of the structural form is emphasized. Thus, novel structures, bridges and ships, which have critical and clear structural forms, have high design times, whereas buildings, which have hidden strength from non-structural parts have lower values. The initial cost, C, includes the actual cost of the structure, design and any insurance provided. Such insurance depends on the chances of failure of the designed structure, p, and hence C = C(p). D is the replacement cost and E is the cost associated with failure due to errors. The chance of this is P and hence E = E(P). A statement of the objective function of costs to be minimized in eqn. (1) is
J(x; p, P)=min[C(x; p)+D(p+P)+E(P).P]
(2)
X
where x are the constrained variables of geometry and weight. The dependence of costs upon x, p and P is clear. However, p is tied to x. Therefore, in the extremum problem of eqn. (2), the x solution to minimize J for prescribed levels of safety, p, and error control, P, is sought. This is satisfactory in a formal sense, but at this time there is not a proven relationship between error control and P. Of concern in eqn. (2) is the error costs, E; these may be very small to the owner if dealt with by insurance premiums. Then, E << C + D and eqn. (2) can be thought of as
J(x; p, P)=min[C(x; p)+D(p+e)]
(3)
X
Decisions on safety and design and construction control establish p and P, leaving x as the optimization variables. If the costs are considered in a more general sense, then E may dominate
TABLE 1 Comparison of the design time per ton of structure of various structural types Type of structure Spacecraft Aircraft Offshore structures: Novel Conventional Bridges and ships Buildings
Engineering man-years 500 10 0.01 0.001 0.0006 0.0002
158
over C and D. Also, the computed probability of failure, p, is often orders of magnitude smaller than the failure incidence [4]. The difference may be accounted for by design-construction-operation errors and hence p << P. These comments suggest that eqn. (3) is often appropriate. Additional to the features in eqn. (2) are costs of the operation of the structure. These, like any long-term costs, have to account for future inflation and interest rates in order to use common initial costs in the objective function. All these pricings, including the elements of eqn. (2), have to be obtained a priori; the usual process is to maintain consistency in assessing each cost alternative. These adjustments amount to exercises in functional and bounded rationality in the senses of Mannheim and Simon.
3. MINIMUM WEIGHT DESIGN The use of weight as an objective function is more direct and simpler than cost. Unfortunately, except in dynamic design, it is a less satisfactory vicarious form. If there are n elements each of weight x(i), then the mathematical program is Y/
rain E x( i )
(4)
i=1
subject to
x(i)
i=1 ton
stress safety constraints displacement constraints stability safety constraints frequency constraints
(5a) (5b) (5c) (5d) (5e)
are prescribed bounds on x. The constraints of eqns. (5b) and (5c) require the Here x and completion of the structural analysis problems K.r--Q
(6)
for each loading and each alternative in the path towards optimality. The constraints of eqns. (5d) and (5e) require the completion of the characteristic value problems
(K-X.M).r=O
(7)
where M is the geometric stiffness matrix for the stability case and is the mass matrix for the frequency one. The resulting stresses and displacements from eqn. (6) are constrained to be no greater than limiting values for all loadings. The actual loads are constrained to be smaller than the factored lowest characteristic loads from eqn. (7) to ensure member stability; the frequencies from eqn. (7) are constrained to avoid specified values. The constraints of eqn. (5a) limit the admissible designs to available member sizes. Search procedures have been developed to determine the optimum variables, x*(i), of the mathematical program of eqns. (4) and (5). These are computationally intensive and require the re-evaluation of eqns. (6) and (7) for each design alternative. Only in fleet or completely novel structures are such search procedures justified. More often the approach is changed to seek the design whereby all members are maximally stressed under at least one loading state. Thus, the optimality criterion of the fully stressed design becomes the surrogate for the minimum weight
159 design. The program search is replaced by a recursion relationship and the constraints of eqns. (5c), (5d) and (5e) are checked on the fully stressed candidate. The mathematical program involving the objective function and constraints may be written as a Lagrangian with the same number of multipliers as the constraints. Such a Lagrangian, L(x, ~), with multipliers, 3'j, is
L(x, g)=f(x)+
Y'~yj{Cj(x)}
(8)
J
with Cj(x) as the homogeneous constraints. A constrained local extremum on x = x* subject to satisfying the K a r u s h - K u h n - T u c k e r conditions.
f(x)
exists at
4. STRUCTURAL SYSTEMS The form of eqn. (1) may apply not only to members but also to structural systems, which are comprised of members with appropriate connectiv!ty and boundary conditions. The constraints for a linear structural system require the solution of eqns. (6) and (7), whether the objective function is for cost or weight. Such a linear structure, with strains restrained to limited elastic response, are applicable to the functional attitude to design. Here failure is specified as a stress, load, displacement or frequency level or range, and such states are obtained by a linear approach. This is usual in space and air structures and for off-shore and ship structures. In buildings and bridges, failure is considered as collapse, and optimization work has been applied to plastic analysis whereby eqn. (6) is replaced by linear programs where the K a r u s h - K u h n - T u c k e r conditions are necessary and sufficient, and where the multipliers, yj, in the dual lead to physical interpretation and a static-kinematic duality [5,6]. This approach has been extended to plastic limit synthesis with weight as the objective function [7]. The computer programming activity in the optimation of structural systems, even when p and P are zero, has been extensive. With adequate resources, such m i n i m u m weight structures can be attained. Interestingly, the iterative process of conventional design, whereby a solution is modified in the light of the proximity to the constraints until a design in which the constraints are satisfied results, has been shown to approach a minimum weight form [8].
5. UNCERTAINTY The statement of eqns. (2) and (3) has uncertainty defined in a frequentist sense evident in p, the probability that the structure will fail when properly designed, constructed and maintained; and P, the probability of errors increasing the failure rate to p + P. Also, the constraints of eqn. (5) have uncertainties imposed in a rational, functional sense inasmuch as the stress and load constraints have a formalized ritual of Level 1 or, at the best, Level 2 methods of assessing structural safety. The significance of these two interpretations of uncertainty in the cost objective function and the safety constraints is deeper than a violation of rational consistency. It is an example of Simon's Principle of Bounded Rationality; either the objectives or the constraints, or both, are incomplete, and the solutions that are forthcoming are not global ones. To illustrate this state consider the objective statement of eqn. (2) where x, p and P are optimization variables. In an
160 unconstrained procedure the design, x, will have minimum costs at safety levels, p and P, which may be unacceptable. The constraints of eqns. (5b) and (5d) are intended to ensure appropriate safety in the collapse sense. These constraints should provide upper limits on P and p, which establish ranges for them in eqn. (2) and hence result in another design, x, different from x, with adequate safety levels, P and p. Such an approach requires that the measures of uncertainties, p and P, are the same in both the objective function and the safety constraints. However, if different measures are used in the cost objective and the safety constraint, then another solution, x, will be found in which no definite statements can be made on the uncertainties p and P. What can be stated are values of p and P on the costs and /3 on the safety. Relationships between /3, the reliability index, and p + P can be made only if p + P have particular forms, such as a normal distribution. Certainly, a satisfactory algorithm exists in this mixed optimization which has functional rationality. However, the meaning is not that of the best of all worlds but an artificial solution of doubtful merit. The expression of the uncertainty as probabilities with a frequentist interpretation in eqns. (2) and (3) is essential in ascertaining costs. The conflict with the usual Bayesian probability in eqns. (5b) and (5d) can be resolved by changing from a cost to a weight objective function. Then objective uncertainty is largely removed, and the subjective uncertainty in eqns. (5b) and (5d) is left. Some concern must still be expressed inasmuch as the minimum weight structure so obtained accounts for realistic uncertainty only an a vicarious /3 form. The use of optimality critera, which is the more normal procedure, to replace cost and weight as objectives suffers from the same absence of realism. However, once again an algorithm with associated functional rationality exists.
6. SATISFICING The Introduction displayed reasons for adopting a satisficing approach to structural design based upon the bounded rationality of Simon and the substantial rationality of Mannheim. Additionally, concern about the statement of costs and the dichotomy in the definitions of failure adopted in the safety constraints and in the objective function tend to suggest a less optimistic and crisp approach to design. The complex nature of a contemporary structure requires sophisticated analysis in the form of eqns. (6) and (7) at each stage in the approach towards a formal optimum. Essentially, the optimization procedure is demanding high precision about a complex system. Zadeh [9] had something to say on this matter in his Principle of Incompatibility: " . . . as the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics." In structural optimization procedures, precision on uncertainty, member location and connectivity, material parameters, and loading and boundary conditions is essential; the analytical tools are an example of precision; the best of all worlds that the solution provides is a precise answer of doubtful significance. There are other, softer ways of proceeding. Roy [10] has provided three approaches relevant to eqn. (1): (1) determine the unique and best solution from x;
161 (2) determine all solutions x ~ X that are "good" and from these consider further to determine those that are equivalently "satisfactory"; (3) determine classes of solutions x ~ X, each of which is equivalent in value, and construct an ordered sequence of these indifference classes. The first approach is optimization; the second two in the end determine alternatives which are of equal value or, more usefully, between which all parties are indifferent. This addresses one aspect of satisficing: the detection of a class of solutions to which indifference between the members of the class exists. The indifference measures need not be precise; they may in fact be vague, but there must be agreement that they evaluate the required structural characteristics of cost, safety and performance in a way that does not allow the actors to prefer one member in an indifference class over another. Such an approach is valid when the constraints are not realistically precise and where the objectives are satisfied by various criteria. A corollary to Zadeh's Principle of Incompatibility has been expressed as: "The closer one looks at a real world problem, the fuzzier becomes its solution." It is this fuzziness or vagueness that can be reflected in approaches (1) and (2) above and which has led to fuzzy decision-making. In this topic, the formalities of mathematical programming of optimization have been mirrored into fuzzy mathematical programming where the objectives and constraints may be fuzzy or where, therefore, the vague membership in a class or set can be dealt with by fuzzy set theory. This has been reviewed recently by Zimmermann [11]. A real complication exists with such an approach inasmuch as the p and P in eqns. (2) and (3) and the fl in the safety constraints are not vague; they are crisp numbers which in the case of p and P are definitely about crisp sets. The fuzzy programming has much the same operational procedures as goal programming and is efficient in obtaining solutions. Essentially, the approach (1) above is being fuzzified. Approaches (2) and (3) rely on ordering and clustering to approach a "best" indifference class. The selection among the alternatives requires further analysis by the comparison of the m goals of the objective function applied to each alternative. Saaty [12] considered this problem from a psychological viewpoint and concluded that reductionism was superior to aggregation. Rather than comparing all alternatives with respect to the composite of goals, he compared alternatives with respect to each goal (A) on a scale of zero to unity. The weighting among the goals involved a characteristic value problem in which the m weights of pairs of goals are determined and assembled into a diagonally inverse m X m matrix. The characteristic vector from the matrix became the weighting vector which then was used to modify the matrix A to obtain a final comparison matrix of the goals for each alternative. The maxi-min alternative provided the decision case. Such an approach, although formal, has the feature of reliablity embedded into it. It appeals to pairwise comparisons and simple algebraic operations to obtain a final sense between the members of the indifference class. It also deals with the multi-objective criteria in crisper ways than when an attemp to compare directly the alternative ranking for all goals. These procedures allow for satisficing among competing alternatives. Another aspect is that the selected alternative must be robust with respect to a range of changes in the parameters, models, actors and uses that are realistically likely to occur. Geometrically, this means that the extremum must be flat and broad in the variable space. Testing for such robustness can be conducted by sensitivity analysis and by indifference to change in the satisficing procedure.
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7. C O M M E N T S
The previous discussion displays a separation between optimizing and satisficing. Optimizing is a procedure based on strong mathematical foundations which allows precise statements to be made in the objective function, constraints and solution. Satisficing is an expectation with little formal trappings which admits of imprecision in the objectives and constraints and robustness in the solution. Applications of optimization in design requires an act of confidence in the procedures which can seldom be supported by the reality of the problem. Applications of satisficing reveal the reality with confidence but are based upon undeveloped procedures. It would appear that some combination of the two would indeed be the best of all possible worlds. However, at the moment optimizing is an act of functional and limited rationality; satisficing has the appearance of global rationality and respects the Principle of Bounded Rationality. Certainly an optimization scheme ignores the Principal of Incompatibility; satisficing may respect the Principle. These comments indicate that future research should attempt to introduce realism into the optimization scheme but, in the process, should anticipate a reduced precision in objective, constraints and solution. The measures for such studies should be the extent that realism co-exists with rigorous analytics. Cooperation between practitioners and researchers seems inevitable if realism is to be tied to what goes on in practice. Indeed, studies that probe the extent that present-day design is a satisficing procedure with convergent and robust methodologies should be examined. Spillers [8] has already given some insight into this approach. Uncertainty will always exist in design of complex structures and Zadeh [9] has suggested that added precision does little or nothing to ameliorate it. The use of probability in a frequentist sense seems to be inevitable in describing costs and should be evident in constraint statements. However, at present a confusion exists whereby constraints are expressed by Bayesian surrogates and costs by implied frequencies. In addition, any move away from precision in the sense of Zadeh [9] calls for measures of imprecision. In this he has advocated the use of fuzzy sets. Some merging must occur between objective and subjective probabilities and fuzzy sets. At the moment there are clear separations in the axioms of probability and fuzzy sets which can be formally studied. However, the meanings of objective and subjective probabilities are dramatically different and lend to confusion when used separately in objectives and constraints.
8. C O N C L U S I O N
An attempt has been made to examine the features of optimizing and satisficing. Optimization is a mathematical activity and can be formally described. The extent that the formalism cannot ape reality has been explored under the umbrella of the principles of Simon [1], Mannheim [3] and Zadeh [9]. These shortcomings suggest another approach to structural design which is easy to appreciate but, as yet, is not established in a formal manner. However, studies can be organized which, in the end, may make use of the formal advantages of optimization and the realism of satisficing. Possible forms of such an hybrid are discussed in the paper.
ACKNOWLEDGEMENT This work was supported by the National Science Foundation, Grant No. ECE 8518155
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